Dirichlet?

Let $S_N$ be the sum of the first $N$ terms of the series in question. We'll show it diverges with the comparison test.

Note that the signs of the terms repeat every five terms. If we concentrate on the first five terms, we just have to observe that $\blue{\frac13>\frac19}$ and $\green{\frac15>\frac{1}{11}}$ :

$S_5=\blue{\frac13}+\green{\frac15}+\frac17-\blue{\frac19}-\green{\frac{1}{11}}>\frac17$

We can do this for every set of five terms; so

$S_{5k}>\frac{1}{7}+\frac{1}{17}+\frac{1}{27}+\cdots+\frac{1}{10k-3}$

Since $\frac{1}{10k-3}>\frac{1}{10k}$ , we have

$S_{5k}>\frac{1}{10}+\frac{1}{20}+\frac{1}{30}+\cdots+\frac{1}{10k}=\frac{1}{10} H_k$

where $H_k$ is the $k^{th}$ harmonic number ; the harmonic series diverges and hence so does the series in question.